The word orbifold was invented in (Thurston 1992), while the original name was VV-manifold (Satake), and was taken in a more restrictive sense, assuming that the actions of finite groups on the charts are always effective. Nowdays we call such orbifolds effective and those which are global quotients by a finite group global quotient orbifolds.

There is also a notion of finite stabilizers in algebraic geometry. A singular variety is called an (algebraic) orbifold if it has only so-called orbifold singularities.

Global quotient orbifolds

In (ALR 07, theorem 1.23) is asserted that every effective orbifold XX (paracompact, Hausdorff) is isomorphic to a global quotient orbifold, specifically to a global quotient of O(n)O(n) (where nn is the dimension of XX) acting on the frame bundle of XX (which is a manifold).

(Co)homology

It has been noticed that the topological invariants of the underlying topological space of an orbifold as a topological space with an orbifold structure are not appropriate, but have to be corrected leading to orbifold Euler characteristics, orbifold cohomology etc. One of the constructions which is useful in this respect is the inertia orbifold (the inertia stack of the original orbifold) which gives rise to “twisted sectors” in Hilbert space of a quantum field theory on the orbifold, and also to twisted sectors in the appropriate cohomology spaces. A further generalization gives multitwisted sectors.

William Thurston, Three-dimensional geometry and topology, preliminary draft, University of Minnesota, Minnesota, (1992) which in completed and revised form is available as his book: The Geometry and Topology of Three-Manifolds; in particular the orbifold discussion is in chapter 13.